Polynomial Rings over Pseudovaluation Rings

Let R be a ring. Let σ be an automorphism of R. We define aσ-divided ring and prove the following.(1) Let R be a commutative pseudovaluation ring such that x∉P for anyP∈Spec(R[x,σ]). Then R[x,σ] is also a pseudovaluation ring.(2) Let R be a σ-divided ring such that x∉P for any P∈Spec(R[x,σ]).Then R[x,σ] is also a σ-divided ring.Let now R be a commutative Noetherian Q-algebra (Q is the field of rationalnumbers). Let δ be a derivation of R. Then we prove the following.(1) Let R be a commutative pseudovaluation ring. Then R[x,δ] is also apseudovaluation ring.(2) Let R be a divided ring. Then R[x,δ] is also a divided ring